Step by step
The question
Question: find the center and the radius of (x + 3)² + (y − 1)² = 8.
1 Compare with the rule
Every circle has this form: (x − h)² + (y − k)² = r².
The center is (h, k).
The radius is r.
2 Find the center
x + 3 is the same as x − (−3).
Taking away a negative is adding.
So h = −3.
Careful: the sign turns.
+ 3 in the bracket means the center has −3.
y − 1 is like y − k.
So k = 1.
So the center is (−3, 1).
3 Find the radius
r² is the number on the right: r² = 8.
r times r is 8.
So r is the square root of 8.
8 is not a square number.
So we keep √8.
That is the exact answer.
With a calculator, √8 is about 2.8, to the nearest tenth.
So the center is (−3, 1) and the radius is √8.
4 Check
Let's check: draw the circle.
Center (−3, 1), radius about 2.8.
Put the center on the grid.
Draw the circle around it.
This point is on the circle: (−1, 3).
It is 2 across and 2 up from the center.
Put x = −1 and y = 3 in.
We get 2² + 2².
4 plus 4 is 8, the number on the right.
Right!
Let's look back.
First: compare with the rule.
Then: find the center.
Then: find the radius.
Last: check.